If I know that this triad is in the key of A major, how can I tell if this a root, first inversion, or second inversion? I know in second inversion, the fifth is usually doubled, but how can I tell that it's a fifth?
In common practice four part harmony the bass note dictates the inversion. So, first identify the chord by the notes that make it:
C#-E-G# --> that's C# min, or iii in A Maj.
So C# is the root of this chord, and it's in the bass, so it's a root chord. If E was in the bass, it would be 1st inversion (C#6); if G# was in the bass it would be 2nd inversion (C#64).
Yes, because it is the bass note, most times the 5th is doubled in 2nd inversion chords. But that also happens with root chords (doubled root) and first inversion (doubled 3rd). So the fact that the bass is doubled does not tells you by itself what kind of inversion it is.
An inversion is just determined by what note of the chord is in the bass. Root position has the root in the bass, first inversion has the third in the bass, second has the fifth in the bass, and third has the 7th in the bass.
Once you know what chord you are looking at, you know what the root, third, and fifth is. I'm not going to give you the answer to this one for obvious reason, but let's look at a similar example in the key of C.
In the example above all three of the chords are the same and the only thing that changes is the bass note thus we have a different inversion for each. In the first example we have the notes G, D, G, B in that order. Since we know that a G major consists of the notes G, B, and D and the root is in the bass we know it's in root position. In the second example we have the notes B, D, G, B in that order. Since we know that a G major consists of the notes G, B, and D and the third is in the bass we know it's in first inversion. In the third example we have the notes D, D, G, B in that order. Since we know that a G major consists of the notes G, B, and D and the fifth is in the bass we know it's in second inversion.
I'd recommend taking a step-by-step approach:
Figure out what the notes are. You can't know what chord it is without knowing what notes there are.
From top to bottom, the notes are: G♯, E, C♯ (twice)
Figure out what chord it is. You can't know what the chord inversion is without knowing what chord it is.
There's only one way to arrange the notes to create the interval of a 3rd between the root and third, and between the third and the fifth: C♯-E-G♯ (C♯ minor). So the chord will use lower-case numerals and won't use the diminished sign (°).
Figure out the inversion.
Root position has the first note (the root) in the bass; the nth inversion has the n+1th note in the bass (or, think of it as shifting the bottom note to the top n times). Since we have a C♯ in the bass and that's the root note of C♯ minor, the chord is in root position. So we won't apply the ⁶ (first inversion) or the ⁶₄ (second inversion) symbols.
Figure out the function of the chord within the key.
C♯ is the 3rd degree of the A major scale, and as per the above we will be using lower case letters with no extra symbols. Final result: iii.